A distributed system is often built on top of an overlay network. Overlay networks enable network topology transparency while, at the same time, can be designed to provide efficient data dissemination, load balancing, and even fault tolerance. They are constructed by defining logical links between nodes creating a node graph. In practice, this is materialized by a Peer Sampling Service (PSS) that provides references to other nodes to communicate with. Depending on the configuration of the PSS, the characteristics of the overlay can be adjusted to cope with application requirements and performance concerns. Unfortunately, overlay efficiency comes at the expense of dependability. To overcome this, one often deploys an application overlay focused on efficiency, along with a safety-net overlay to ensure dependability. However, this approach results in significant resource waste since safety-net overlays are seldom used. In this paper, we focus on safety-net overlay networks and propose an adaptable mechanism to minimize resource usage while maintaining dependability guarantees. In detail, we consider a random overlay network, known to be highly dependable, and propose BuzzPSS, a new Peer Sampling Service that is able to autonomously fine-tune its resource consumption usage according to the observed system stability. When the system is stable and connectivity is not at risk, BuzzPSS autonomously changes its behavior to save resources. Alongside, it is also able to detect system instability and act accordingly to guarantee that the overlay remains operational. Through an experimental evaluation, we show that BuzzPSS is able to autonomously adapt to the system stability levels, consuming up to 6x less resources than a static approach.